To be equivalent, it would have to be the US saying that Canadians and Mexicans were "really" just "states" of America (as Beijing insists that Cantonese is "really" just a "dialect" of Chinese) and intentionally attempting to marginalize Canadian / Mexican distinctives.
I don't think I'd mind calling Mandarin "Chinese" if they didn't then insist that Cantonese is also "Chinese", use that to gaslight Cantonese speakers into thinking that their spoken language wasn't a real language, but "full of slang", and that their written language needed to be closer to Mandarin to be "correct Chinese".
Not in my experience (lifelong American). I'm not aware of any project to make people think all of North and South America are, in fact, rightly part of the U.S.[1]
The only slightly similar thing I can recall is an impression I formed in the 1980s: the global reach of Hollywood's TV and films gave non-American's a very skewed view of life in America.
[1] other than America's Westward expansion, and as far south as the Rio Grande river. But IIUC those were militaristic expansions, not mere cultural imperialism.
Apologies, I might not have been clear.
The annoyance from Mexican friends is that Americans (apparently) often think that Canada and the US are "North America", and Mexico and southward are "South America". Or that "Central America" is a continent, and that Mexico is part of that continent.
Boiled down, the erroneous belief is that Canada and the US are part of "North America", and Mexico is part of something else.
I can't evaluate this on its merits. I have a good memory for geography, and zero confusion on this matter. Not all Americans share these traits (there is plenty of evidence for this!).
And I am not a Mexican who has ever fielded the situation with a confused American, but I take them at their word.
And, we are taught a lot of things in school that are not retained. Some 50% of American high school graduates can't find Florida on a map.
But I wonder if this isn't just the US version of your Mandarin step 4. The US has dominated the global cultural conversation and the "other americans" have given up or never thought to challenge it.
...
Distantly related: I have spoken to Mexicans who are annoyed that Americans frequently (wrongly) think that only Canada and the US are "North America".